You can edit your material in a Project that matches it, then you can Export/Share/Publish in about any size (I would keep the Aspect Ratio the same, and divide by factors of 8, to help with quality) that you want.
The File Size is determined by two factors: Duration (RunTime) and Bit-Rate. The Frame Size will not have much impact.
Now, Bit-Rate is tied directly to File Size, and also Quality. I would investigate lowering the Bit-Rate in your output file, checking the quality. If you run out of "quality," and the File Size is still too large, you can explore various formats and CODEC's. Some, like DivX, are designed to work with a lower Bit-Rate, but still with good quality for streaming. MS's WMV can also yield pretty good quality at lower Bit-Rates.
The limitations of those two are that with DivX, the viewer must have that CODEC, or perhaps use a player like VLC Player, or MediaPlayer Classic HC (which have their own sets of CODEC's). The DivX CODEC is very common, however, so that might not be a big deal. With WMV, a Mac-user will need a utility/player, like Flip4Mac.
What version of Premiere Elements are you using and on what computer operating system are you running it?
There are many factors that impact your video file size including frame size, frame rate, as well as durationXbitrate, video compression and audio compression to name a few.
Do you have more than the 2 GB free space from Dropbox? And, what is the file size and duration and bitrate of the video clip that you are importing into Premiere Elements for editing? How much of a drop in file size are you looking for between the import into and export from Premiere Elements? What are the requirements for max file size for a single upload to Dropbox and what are the Dropbox supported video and audio formats?
Given that you have 1920 x 1080 @ 24 progressive frames per second footage and assumed that you have imported into Premiere Elements X using the project preset
NTSC or PAL
and you are ready to export to a file for upload to Drop Box.
If you are using Premiere Elements 11 Windows consider for starter:
and explore the use of the presets
a. MP4 - H.264 1920 x 1080p24, as is, and with settings adjustments under the Advanced Button/Video Tab of the preset.
b. MP4 - 720p24
The Windows Media export will probably give you the smaller file size if needed. Export settings under the Advanced Button/Video tab of a preset should be looked even if the Presets: default does not look applicable to your situation.
Please review and then let us know how we can help further.